Prof. Stefano Bianchini (SISSA, Trieste),
who will talk about
LAGRANGIAN REPRESENTATION FOR CONSERVATION LAWS.
Prof. Bianchini has been recently awarded the Magenes Prize of Unione Matematica Italiana. The Magenes Prize is awarded every 4 years to a mathematician that obtained relevant results in the theoretical or numerical analysis of differential models and their applications.
The Colloquium Magenes is also the opening talk of the Workshop on Recent Trends in PDEs
Everybody is invited.
The Colloquium Magenes is jointly organized by the Department of Mathematics and by IMATI-CNR to honor the memory of Prof. Enrico Magenes. It consists of a conference given every year by a leading expert in the field of numerical or theoretical analysis and its applications.
The use of characteristics to represent smooth solutions in quasilinear first order systems is textbook classic, and it is also well known that after the so called gradient catastrophe, this representation fails because the solution becomes discontinuous and there is no canonical way to continue the representation. For linear transport equations, instead, even if the solution is very weak (say a measure) and the vector field is only locally integrable, a representation in terms of superposition of characteristics is the base of important progresses regarding uniqueness and existence of a more regular flow.
In this talk I will show how the method of characteristic can be extended to scalar equations and hyperbolic systems of conservation laws, yielding a new representation of the solution, the Lagrangian representation. I will address in particular the following points:
- questions where the Lagrangian representation arises naturally;
- the Lagrangian representation as a continuous wave tracing;
- fine description of L^\infty-solutions for scalar equations and BV-solutions for systems.